How to connect end and beginning of a "ring" with a cutout?
Toggling Cyclic on your curve should not (and does not) affect the shape of existing curve, especially since it is poly type. That could only happen if there were any Automatic control points in it.

If it truly is a 2D curve then press the 2D button to ensure it actually is bidimensional.

In your case you will notice it will ruin your geometry because it is not correctly oriented in local space. Enter edit mode, rotate it 90 degrees in the X axis before making it 2D.

You can finally press Alt+C in edit mode to make it Cyclic.